1. Recursion

2. Recursion Recursion is a technique widely used in programming
In fact some programming languages have no loops, only recursion
Recursion is a key element to advanced studies such as computational intelligence
Beginning programmers usually groan at recursion… real programmers love it!

3. Recursion in Java A method is recursive if it calls itself.
There are two key things in recursion
Base step: ends the recursion
Recursive step: breaks down the problem to smaller sub problems
Before you code anything, you need to find those two steps… otherwise you may not understand the problem properly

4. Example Here is a simple recursive java function for called countdown:
public void countDown( int value ) {
if( value == 0 )
System.out.println( “Blast Off!!!” );//base case, no recursion
else
System.out.println( value );
countDown( value – 1);//recursive call of small problem }
LETS TRACE countDown( 3 ) BY HAND

5. Why Recursion?
Some algorithms are exceedingly complex as a loop, yet very elegant in a recursive form.
Easier to understand  less “bugs” = better programs
Let’s take a look at Paint for a minute…

6. The Flood Fill Algorithm
The recursion is tricky to follow, but look at how simple the code is!
public void floodFill( int x, int y, Colour clicked, Colour change ) {
if( outOfBounds(x,y) || getColour(x,y) == change || getColour(x,y) != clicked )
return;//nothing to do – base case
setColour(x,y,change);//colour is the same as was clicked, change it
//Recursive call on neighbours
floodFill( x + 1, y, clicked, colour );
floodFill( x, y + 1, clicked, colour );
floodFill( x - 1, y, clicked, colour );
floodFill( x, y - 1, clicked, colour ); }

7. Getting the hang of it? Let’s try a couple together Sum of n numbers
Base step
Recursive step
Sum of n numbers
Ex = 15
Factorials
0! = 1
3! = 3 x 2 x 1 = 3 x 2!
100! = 100x99x98x..x3x2x1 = 100 x 99!

8. Still Need Practice? Here are two other famous recursive examples:
Fibonacci numbers
1st number is 1, 2nd number is 1
Every number after 2 is the sum of the previous two: 1,1,2,3,5,8,13,…
Euclid’s GCF (from ~300BC)
Take the larger of the two and divide it by the smaller
If the remainder is zero, return the smaller number
Otherwise take the GCF of smaller and remainder
Example let’s try 60 and 105

9. Pascal’s Triangle 1 2 3 4 6 …

10. Pascal’s Triangle Cont’
What’s the recursive pattern in Pascal’s triangle?
Hint: Number each row and column
What is/are the base case(s)?
What is the recursive case?

11. Pascal’s Triangle – Programmer’s View1 2 3 4 6 5 10 …

12. Performance: Loops vs. Recursion
Why not always use recursion if it makes our code easier to understand?
The key reason is performance
Recursion can quickly overflow the computer’s memory
Each method call has its own “stack” in memory when it is called (holds variables, loops, etc.)
When you make a recursive call, another copy of the method is stored in memory to preserve the state when it returns.
With loops, the variables are reused
Last I had heard, Sun’s java compiler will translate tail recursion into a loop for you
Tail recursion means the recursive call is the last piece of code in the method (nothing is computed after it returns)

13. Recursion in Memory Consider a call to factorial( 3 ):
factorial(3)  3 * factorial(2)
 2 * factorial(1)
 1 * factorial(0) 1
Recursion ends, return to last caller:
1 * 1 = 1
 2 * 1 = 2
 3 * 2 = 6

14. Recursion in Memory With a loop it would look like this:
public int factorial( int n ) {
int fac = 1;
for( int counter = 2; counter <= n; counter++ )
fac *= counter;
return fac; }
A call to factorial( 3 ) is now just:
1 * 2 * 3 = 6

15. FIbonacci
Since this problem is doubly recursive things get “bad” quickly with recursion.
Let’s hand trace fib(4) on the board

16. Exercises Create a recursive method to calculate the sum of squares:
For example, sumSquares( 3 )
Is = 14

17. Exercises
Create a recursive method to calculate positive integer exponents
Ex.
findExponent( 3, 2 )
Returns 9

18. Exercises
Create a recursive method to determine if the values in an array are increasing
You need an extra parameter telling you which index to start testing from
isIncr( int[] data, int start )
isIncr( {1,3,5,9},0 ) returns true
isIncr( {3,1,2,4},0 ) returns false

19. Exercises Create a recursive method to sum the values in an array:
sum( int[] data, int start )
sum( {4,2,5}, 0 )
Returns 11

20. Exercises
Create a recursive method called reverse that reverses the elements in an array
Reverse( int[] data )
Reverse( {1,3,5,2,7 } )
{7,2,5,3,1}

21. Exercises – Reverse Cont’
Sometimes when we are using recursion, we will write a helper method to hold more parameters.
In the previous example to reverse an array you should write a helper method called:
ReverseHelper( int[] data, int start, int end )
When you call Reverse( int[] data ) use the helper method as follows (it does all the work)
ReverseHelper( data, 0, data.length)

22. Exercises
Create a recursive method to determine whether or not a word is a palindrome or not.
A palindrome is a word spelled the same forwards and backwards
For example Ogopogo
public boolean isPalidrome(String word)

23. Exercises
To convert an integer to base 2 (binary) here is an algorithm:
public String toBinary( int n )
If the number is zero, return 0
Otherwise,
Compute the quotient and remainder when dividing by 2
The binary value of the number given is the binary value of the quotient followed by the remainder

24. Binary Numbers Ex. Binary value of 13: Calculation Quotient Remainder
13/ 6/ 3/ 1/
0 return 0
13 = in binary